Seeking a network characterization of Corson compacta

Ziqin Feng

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 4, page 513-521
  • ISSN: 0010-2628

Abstract

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We say that a collection 𝒜 of subsets of X has property ( C C ) if there is a set D and point-countable collections 𝒞 of closed subsets of X such that for any A ∈ 𝒜 there is a finite subcollection ℱ of 𝒞 such that A = D ∖ ⋃ ℱ . Then we prove that any compact space is Corson if and only if it has a point- σ - ( C C ) base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by B. Cascales and J. Orihuela and as in “Network characterization of Gul’ko compact spaces and their relatives” (2004) by F. Garcia, L. Oncina, J. Orihuela, which asked whether there is a network characterization of the class of Corson compacta.

How to cite

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Feng, Ziqin. "Seeking a network characterization of Corson compacta." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 513-521. <http://eudml.org/doc/298007>.

@article{Feng2021,
abstract = {We say that a collection $\mathcal \{A\}$ of subsets of $X$ has property $(CC)$ if there is a set $D$ and point-countable collections $\mathcal \{C\}$ of closed subsets of $X$ such that for any $A\in \mathcal \{A\}$ there is a finite subcollection $\mathcal \{F\}$ of $\mathcal \{C\}$ such that $A=D\setminus \bigcup \mathcal \{F\}$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma $-$(CC)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by B. Cascales and J. Orihuela and as in “Network characterization of Gul’ko compact spaces and their relatives” (2004) by F. Garcia, L. Oncina, J. Orihuela, which asked whether there is a network characterization of the class of Corson compacta.},
author = {Feng, Ziqin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Corson compacta; point network; condition (F); almost subbase; additively $\aleph _0$-Noetherian},
language = {eng},
number = {4},
pages = {513-521},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Seeking a network characterization of Corson compacta},
url = {http://eudml.org/doc/298007},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Feng, Ziqin
TI - Seeking a network characterization of Corson compacta
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 513
EP - 521
AB - We say that a collection $\mathcal {A}$ of subsets of $X$ has property $(CC)$ if there is a set $D$ and point-countable collections $\mathcal {C}$ of closed subsets of $X$ such that for any $A\in \mathcal {A}$ there is a finite subcollection $\mathcal {F}$ of $\mathcal {C}$ such that $A=D\setminus \bigcup \mathcal {F}$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma $-$(CC)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis” (2014) by B. Cascales and J. Orihuela and as in “Network characterization of Gul’ko compact spaces and their relatives” (2004) by F. Garcia, L. Oncina, J. Orihuela, which asked whether there is a network characterization of the class of Corson compacta.
LA - eng
KW - Corson compacta; point network; condition (F); almost subbase; additively $\aleph _0$-Noetherian
UR - http://eudml.org/doc/298007
ER -

References

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